# Generalized Schoenflies - formalizing step in proof?

[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]

I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, which is a specific case of the generalized Schoenflies theorem:

Every smoothly embedded $$S^2\subset \mathbb{R}^3$$ bounds a smooth 3-ball.

The proof seems to rely on intuition for these low-dimensional arguments, which I find disconcerting, because I have not yet developed that intuition, so I am trying to give actual formal proofs for the statements in Hatcher's proof.

The proof begins with a generic smoothly embedded closed surface $$S\subset\mathbb{R}^3$$. I have been able to prove that I can isotope $$S$$ so that projection on the last coordinate $$\pi:\mathbb{R}^3\rightarrow\mathbb{R}$$ is a Morse function on $$S$$. Hatcher then argues that if $$t$$ is a regular value for $$\pi$$, then $$\pi^{-1}(t)\cap S$$ is a finite collection of circles.

The proof continues by taking an innermost circle $$C\subset \pi^{-1}(t)\cap S$$, which by 2-dimensional Schoenflies bounds a disk $$D$$, and $$D\cap S=\partial D=C$$. Hatcher then uses surgery to cut away a neighborhood of $$C$$ in $$S$$, and cap the cuts with two disks.

This last part is what I want to formalize. It seems we are finding a small-enough tubular neighborhood $$C\times(-\epsilon,\epsilon)\subset S$$, and then removing that, leaving $$S_-=C\times\{-\epsilon\}$$ and $$S_+=C\times\{\epsilon\}$$. Again by 2-dimensional Schoenflies, these bound disks $$D_-$$ and $$D_+$$. What I don't get is: why are $$S_-$$ and $$S_+$$ still innermost? Or, put differently, why is $$D_-\cap S=S-$$ (and similarly for $$S_+$$)?

Intuitively, this seems obvious, and it seems like some sort of "continuity" argument would work, but I cannot figure out how to make this formal. I tried proving that in fact all the disks, "stacked" together for the tubular neighborhood, gave a smooth $$D\times [-\epsilon,\epsilon]$$, but again I find it hard to make topological arguments when one step of the construction is "apply Schoenflies to get a disk". In particular, I can't prove the projection of this "solid neighborhood" to $$D$$ is continuous.

Does anyone know how to formalize this? Or, even better, a reference where this type of surgery is discussed? I checked a few places, but only found surgery on a single manifold, not the type discussed here, where we're surgering an embedded submanifold in some ambient manifold.

If $$t$$ is a regular value, then it is a property of Morse functions that there is some small open neighborhood $$U$$ of $$t$$ in $$\mathbb{R}$$ such that $$u$$ is also a regular value for all $$u\in U$$. In particular, we can take $$U=(t-\delta,t+\delta)$$ for some $$\delta>0$$. But then $$\pi^{-1}(U)\cap S\cong(\pi^{-1}(t)\cap S)\times U$$, ie. the surface is a product between any two successive critical levels. Now simply choose $$\epsilon<\delta$$. Then $$S_-=(C\times U)\cap\pi^{-1}(t-\epsilon)$$ and $$S_+=(C\times U)\cap\pi^{-1}(t+\epsilon)$$ will be innermost since $$C$$ is innermost.
• That cannot happen because there is also a product $D\times U$. – Josh Howie Feb 3 at 22:21